arXiv subjects

Priya Subramanian

Publications and source records attributed to Priya Subramanian.

12 recordsLinked to original sources

arXiv2026

Resonance structure of a periodically forced delay differential equation model for the El Ni\~no--Southern Oscillation

We study resonance phenomena in the periodically forced Suarez--Schopf delay differential equation, which is a conceptual climate model for the El Ni\~no--Southern Oscillation (ENSO). The system serves as a prototypical forced delayed-action oscillator whose self-sustained oscillations, when subjected to periodic forcing, give rise to attracting invariant tori. We provide a comprehensive bifurcation analysis of both the unforced and the forced model; for the latter, we propose a method to compute the rotation number of normally hyperbolic attracting invariant tori. With it we show that resonance tongues in parameter space are organized by critical points of the graph of the rotation number, both along torus bifurcation curves and within the region of invariant tori. We also show that the resonance structure repeats for large delays, which constitutes a reappearance mechanism not previously reported in the literature. Furthermore, depending on the feedback strength, we find bistability between period-one orbits and invariant tori. This regime involves non-classical bifurcation sequences, including `saddle-node' and `gluing' bifurcations of tori.

math.DS↗

arXiv2026

Devil's terraces: determining the organization of resonance tongues in a periodically forced dynamical system

In periodically forced dynamical systems, resonance tongues are open regions of a parameter plane in which the dynamics on an invariant torus locks to a stable periodic orbit. While individual resonance tongues are well understood, the principles governing their global arrangement remain largely unexplored. We develop a topological framework, grounded in applied topology and Morse theory, whose central object is the two-dimensional resonance surface, defined as the graph of the rotation number $\rho$ over a parameter plane. Within this framework, resonance tongues appear as terraces of the resonance surface at rational values of $\rho$, and their global arrangement is determined by the singularities of this surface. Resolving the resonance surface requires the accurate computation of $\rho$, and we present an algorithm that does so efficiently and at high resolution. As a specific example, we examine a periodically forced model of vertical mixing in the North Atlantic, a process relevant to the Atlantic Meridional Overturning Circulation, and study how its resonance surface changes under variation of a third parameter. We identify six distinct resonance-tongue arrangements and show that the resonance transitions between them are due to changes in the number and type of singularities on the boundary of the resonance surface.

math.DS↗

arXiv2026

Regular and chaotic Welander oscillations in a four-dimensional conceptual model for the Atlantic Meridional Overturning Circulation

The Atlantic Meridional Overturning Circulation (AMOC) is a key component of the Earth's climate. Evidence indicates a twentieth-century weakening, and enhanced freshwater input to the subpolar North Atlantic may further reduce overturning strength. We present and study a conceptual four-dimensional, single-hemisphere box model with three compartments: a tropical surface box, a subpolar surface box, and a large deep-water box. Advective exchange couples the surface boxes and vertical exchange with the deep ocean is represented by a smooth convective-adjustment scheme. A comprehensive bifurcation analysis reveals an equilibrium structure with up to four coexisting overturning states, together with regimes of bistability and tristability. We identify families of periodic solutions and chaotic attractors with a clear timescale separation: a millennial oscillation is modulated by faster decadal-to-centennial variability arising from episodic shutdowns of subpolar convection. As prescribed freshwater fluxes increase, shutdown events become more frequent and the background overturning weakens. Additionally, for certain values of freshwater influx, the dynamics become chaotic, producing an irregular on-off switching of convection.

math.DS↗

arXiv2022

Forced symmetry breaking as a mechanism for rogue bursts in a dissipative nonlinear dynamical lattice

We propose an alternative to the standard mechanisms for the formation of rogue waves in a non-conservative, nonlinear lattice dynamical system. We consider an ODE system that features regular periodic bursting arising from forced symmetry breaking. We then connect such potentially exploding units via a diffusive lattice coupling and investigate the resulting spatio-temporal dynamics for different types of initial conditions (localized or extended). We find that in both cases, particular oscillators undergo extremely fast and large amplitude excursions, resembling a rogue wave burst. Furthermore, the probability distribution of different amplitudes exhibits bimodality, with peaks at both vanishing and very large amplitude. While this phenomenology arises over a range of coupling strengths, for large values thereof the system eventually synchronizes and the above phenomenology is suppressed. We use both distributed (such as a synchronization order parameter) and individual oscillator diagnostics to monitor the dynamics and identify potential precursors to large amplitude excursions. We also examine similar behavior with amplitude-dependent diffusive coupling.

nlin.PS↗

arXiv2021

Density Distribution in Soft Matter Crystals and Quasicrystals

The density distribution in solids is often represented as a sum of Gaussian peaks (or similar functions) centred on lattice sites or via a Fourier sum. Here, we argue that representing instead the $\mathit{logarithm}$ of the density distribution via a Fourier sum is better. We show that truncating such a representation after only a few terms can be highly accurate for soft matter crystals. For quasicrystals, this sum does not truncate so easily, nonetheless, representing the density profile in this way is still of great use, enabling us to calculate the phase diagram for a 3-dimensional quasicrystal forming system using an accurate non-local density functional theory.

cond-mat.soft↗

arXiv2020

Snaking without subcriticality: grain boundaries as non-topological defects

Non-topological defects such as grain boundaries abound in pattern forming systems, arising from local variations of pattern properties such as amplitude, wavelength, orientation, etc. We introduce the idea of treating such non-topological defects as spatially localised structures that are embedded in a background pattern, instead of treating them in an amplitude-phase decomposition. Using the two-dimensional quadratic-cubic Swift--Hohenberg equation as an example we obtain fully nonlinear equilibria that contain grain boundaries which are closed curves containing multiple penta-hepta defects separating regions of hexagons with different orientations. These states arise from local orientation mismatch between two stable hexagon patterns, one of which forms the localised grain and the other its background, and do not require a subcritical bifurcation connecting them. Multiple robust isolas that span a wide range of parameters are obtained even in the absence of a unique Maxwell point, underlining the importance of retaining pinning when analysing patterns with defects, an effect omitted from the amplitude-phase description.

nlin.PS↗

arXiv2020

Spatial localisation beyond steady states in the neighbourhood of the Takens--Bogdanov bifurcation

The coincidence of a pitchfork and Hopf bifurcation at a Takens-Bogdanov (TB) bifurcation occurs in many physical systems such as double-diffusive convection, binary convection and magnetoconvection. Analysis of the associated normal form, in one dimension with periodic boundary condition, shows the existence of steady patterns, standing waves, modulated waves and travelling waves, where the values of coefficients of the terms in the normal form classify all possible different bifurcation scenarios in the neighbourhood of the TB bifurcation (Dangelmayr & Knobloch, 1987). In this work we develop a new and simple pattern-forming PDE model, based on the Swift-Hohenberg equation, adapted to have the TB normal form at onset, which allows us to explore the dynamics in a wide range of bifurcation scenarios, including in domains much wider than the lengthscale of the pattern. We identify two bifurcation scenarios in which coexistence between different types of solutions is indicated from the analysis of the normal form equation. In these scenarios, we look for spatially localised solutions by examining pattern formation in wide domains. We recover two types of localised states, that of a localised steady state in the background of the trivial state and that of a spatially localised travelling wave in the background of the trivial state which have previously been observed in other systems. Additionally, we identify two new types of spatially localised states: that of a localised steady state in a modulated wave background and that of a localised travelling wave in a steady state background. The PDE model is easy to solve numerically in large domains and so will allow further investigation of pattern formation with a TB bifurcation in one or more dimensions and the exploration of a range of background and foreground pattern combinations beyond steady states.

nlin.PS↗

arXiv2020

Exploring Critical Points of Energy Landscapes: From Low-Dimensional Examples to Phase Field Crystal PDEs

In the present work we explore the application of a few root-finding methods to a series of prototypical examples. The methods we consider include: (a) the so-called continuous-time Nesterov (CTN) flow method; (b) a variant thereof referred to as the squared-operator method (SOM); and (c) the the joint action of each of the above two methods with the so-called deflation method. More traditional methods such as Newton's method (and its variant with deflation) are also brought to bear. Our toy examples start with a naive one degree-of-freedom (dof) system to provide the lay of the land. Subsequently, we turn to a 2-dof system that is motivated by the reduction of an infinite-dimensional, phase field crystal (PFC) model of soft matter crystallisation. Once the landscape of the 2-dof system has been elucidated, we turn to the full PDE model and illustrate how the insights of the low-dimensional examples lead to novel solutions at the PDE level that are of relevance and interest to the full framework of soft matter crystallization.

nlin.PS↗

arXiv2020

Spatiotemporal chaos and quasipatterns in coupled reaction-diffusion systems

In coupled reaction-diffusion systems, modes with two different length scales can interact to produce a wide variety of spatiotemporal patterns. Three-wave interactions between these modes can explain the occurrence of spatially complex steady patterns and time-varying states including spatiotemporal chaos. The interactions can take the form of two short waves with different orientations interacting with one long wave, or vice-versa. We investigate the role of such three-wave interactions in a coupled Brusselator system. As well as finding simple steady patterns when the waves reinforce each other, we can also find spatially complex but steady patterns, including quasipatterns. When the waves compete with each other, time varying states such as spatiotemporal chaos are also possible. The signs of the quadratic coefficients in three-wave interaction equations distinguish between these two cases. By manipulating parameters of the chemical model, the formation of these various states can be encouraged, as we confirm through extensive numerical simulation. Our arguments allow us to predict when spatiotemporal chaos might be found: standard nonlinear methods fail in this case. The arguments are quite general and apply to a wide class of pattern-forming systems, including the Faraday wave experiment.

nlin.PS↗

arXiv2020

Invariant states in inclined layer convection. Part 2. Bifurcations and connections between branches of invariant states

Convection in a layer inclined against gravity is a thermally driven non-equilibrium system, in which both buoyancy and shear forces drive spatio-temporally complex flow. As a function of the strength of thermal driving and the angle of inclination, a multitude of convection patterns is observed in experiments and numerical simulations. Several observed patterns have been linked to exact invariant states of the fully nonlinear 3D Oberbeck-Boussinesq equations. These exact equilibria, traveling waves and periodic orbits reside in state space and, depending on their stability properties, are transiently visited by the dynamics or act as attractors. To explain the dependence of observed convection patterns on control parameters, we study the parameter dependence of the state space structure. Specifically, we identify the bifurcations that modify the existence, stability and connectivity of invariant states. We numerically continue exact invariant states underlying spatially periodic convection patterns at Pr=1.07 under changing control parameters for temperature difference between the walls and inclination angle. The resulting state branches cover various inclinations from horizontal layer convection to vertical layer convection and beyond. The collection of all computed branches represents an extensive bifurcation network connecting 16 different invariant states across control parameter values. Individual bifurcation structures are discussed in detail and related to the observed complex dynamics of individual convection patterns. Together, the bifurcations and associated state branches indicate at what control parameter values which invariant states coexist. This provides a nonlinear framework to explain the multitude of complex flow dynamics arising in inclined layer convection.

nlin.PS↗

arXiv2019

Deriving phase field crystal theory from dynamical density functional theory: consequences of the approximations

Phase field crystal (PFC) theory, extensively used for modelling the structure of solids, can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Standard derivations neglect a term of form $\nabla\cdot[n\nabla L n]$, where $n$ is the scaled density profile and $L$ is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in $n$. However, the consequences of doing this are the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and other periodic structures are erroneously predicted to be thermodynamic equilibria. A second approximation is to replace $L$ by a gradient expansion. This leads to the possibility of solutions failing to exist above a certain value of the average density. We illustrate these conclusions with a simple model two-dimensional fluid. The consequences of the PFC approximations are that the phase diagram is both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal. Thus, although PFC models are successful as phenomenological models of crystallisation, we find it impossible to derive the PFC model as an accurate approximation to DDFT, without introducing spurious artefacts. However, making a simple one-mode approximation for the logarithm of the density distribution is surprisingly accurate, which gives a tantalising hint that accurate PFC-type theories may instead be derived for the field $\log(ρ(x))$, rather than for the density profile itself.

cond-mat.soft↗

arXiv2016

Spatio-temporal Patterns in Inclined Layer Convection

This paper reports on a theoretical analysis of the rich variety of spatio-temporal patterns observed recently in inclined layer convection at medium Prandtl number when varying the inclination angle $γ$ and the Rayleigh number $R$. The present numerical investigation of the inclined layer convection system is based on the standard Oberbeck-Boussinesq equations. The patterns are shown to originate from a complicated competition of buoyancy-driven and shear-flow driven pattern forming mechanisms. The former are expressed as \rm{longitudinal} convection rolls with their axes oriented parallel to the incline, the latter as perpendicular \rm{transverse} rolls. Along with conventional methods to study roll patterns and their stability, we employ direct numerical simulations in large spatial domains, comparable with the experimental ones. As a result, we determine the phase diagram of the characteristic complex 3D convection patterns above onset of convection in the $γ-R$ plane, and find that it compares very well with the experiments. In particular we demonstrate that interactions of specific Fourier modes, characterized by a resonant interaction of their wavevectors in the layer plane, are key to understanding the pattern morphologies.

physics.flu-dyn↗